Overview
Have you ever thought “If an electron induces a magnetic field when it moves, and magnetic fields contain
energy, where does the energy to create this field come from?”
It comes from the force that accelerated the electron from rest - the work done in accelerating it (as the
integral of the force over the distance) directly creates the energy of the induced field. Another way of saying this is that
part of the electron’s inertia comes from supplying energy to the induced magnetic field.
We do not know the fine detail of the electron’s electric field structure, only the way it falls
off as the square of the distance when we are some way away from the electron (in terms of the electron size). So it is easier
to deal with the inertia of simple electromagnetic fields. The equations are straightforward and derived for Newtonian velocities,
using source equations to be found in any first-year physics textbook.
Electrostatic inertia - the basics
If a point in space has a vector electrostatic field of strength ‘X’ volts/meter,
its energy density over space ‘dEE/dS’ joules/meter3 is given by...
If this field is moving at a velocity vector ‘v’ relative to the observer it induces
a vector magnetostatic field of induction ‘B’...
(Note that the symbol ‘x’ refers to the vector cross-product, so (X x v) is the
vector X’s cross-product term with v, vectors all being in a bold typeface.)
This induced magnetostatic field has an energy density ‘dEM/dS’ of...
The cross-product ‘(X x v)’ is zero where the vectors ‘X’
and ‘v’ are parallel to each other, and a maximum where they are orthogonal. At this maximum...
...so the ratio of the induced magnetostatic field energy density to the primary electrostatic field energy
density is simply v2/c2 at this maximum.
If the orientation between the electrostatic field and the velocity vector is other than the optimum of 90
degrees, the ratio of induced magnetic energy to the rest electrostatic energy is v2.cos2/c2.
The “cos” term refers to the cosine of the angle between the velocity vector and the orientation of the primary
electrostatic field lines.
Electrostatic inertia - problem in Relativity! Click here to skip the heavy stuff!
Can we apply the maths to the electron on the assumption that the electron is a pure radial electric field?
Because of its spherical symmetry the electron can be dealt with by three orthogonal vectors - one along
the direction of motion, and the other two normal to the direction of motion, with the rest energy of the electric equally
distributed between them. The 1/3rd along the direction of motion induces no magnetic energy, while the other two
vector directions induce 1/3rd of v2/c2 each, so the total induced energy is 2/3rd
of v2/c2.
Now the actual kinetic energy of an electron at Newtonian speeds is one half of v2/c2,
so this result is 4/3rd of what we might expect. This is famous as the “4/3 Problem”. In 1922 Enrico
Fermi, in the paper “Il Nuovo Cimento”, derived this result relativistically (the maths here is the Classical
solution), but showed nothing new - after all the relativistic approach must reduce to the Classical approach at sub-relativistic
speeds. Feynman’s “Lectures on Physics” showed how this 4/3 result violated relativity. The conclusion often
made is that the electron could not be purely electromagnetic. However the calculation was made for a radial electrostatic
field (albeit on the working assumption that the electron could be modelled by this) and hence the proper conclusion
is that a radial electrostatic field violates relativity. By geometrical inference all electrostatic field structures violate
relativity. We cannot suspect the laws of induction, which are well-proven - electric motors work because induced fields containing
energy, nor can we suspect the laws of relativity, which are again well proven, but there is little doubt the two are in conflict.
There is no mechanism for an electrostatic field to propagate like an electromagnetic wave yet it has an effect at a distance.
Either the electrostatic field must convert to an electromagnetic wave and then magically convert back when it sees another
electrostatic field it can interact with, or else no energy is involved in its propagation implying that the speed of propagation
of an electrostatic wave is either zero or infinite - the only possible solutions for this condition for any propagating
field. The former is obviously disallowed by experiment, but no-one has ever worked out the speed of propagation of an elctrostatic
wave from first principles as has been done for eletromagnetic waves to check the latter. Bear in mind that electromagnetic
waves propagate by a falling magnetic field creating a rising electric field, and vice-versa, while moving electrostatic fields
simply generate magnetic fields in situ. A electric dipole generates electrostatic and electromagnetic fields; near the dipole
the electrostatic fields dominate, but the field falls off at 1/r3 while the electromagnetic fields fall off only
as 1/r2, so more then a wavelength or more away from the aerial the electromagnetic waves dominate.
There is an conflict between the scientists who work with relativistic particles and photons, and those who
work with motion-induced induction, and neither group is interested in resolving it.
Magnetostatic inertia
For completeness it is worth looking at the induction of a moving magnetostatic field.
If a point in space has a magnetostatic field of induction ‘B’, energy density...
which is moving at a velocity ‘v’ relative to the observer, it induces an electrostatic
vector field of strength ‘X’...
X = (B x v)
...with an energy density...
The induced field is zero when ‘B’ and ‘v’ are parallel,
and reaches its maximum when they are orthogonal to each other. At this maximum...
...so again the ratio of the induced electrostatic field energy density to the primary magnetostatic field
energy density is simply v2/c2 at this maximum.
If the orientation between the magnetostatic field and the velocity vector is other than the optimum of 90
degrees, the ratio of induced electrostatic energy to the rest magnetostatic energy is v2.cos2/c2.
The “cos” term refers to the cosine of the angle between the velocity vector and the orientation of the primary
magnetostatic field lines.
Conclusion
Electrostatic and magnetostatic fields exhibit inertial properties, but it is fair to say that there are
issues awaiting resolution.
Appendix Application to Inductance
The phenomenon of inductance is clear proof that the electric field of a moving electron has inertia.
Electrostatic inertia is associated with the concept of inductance, used in electric circuits; creating the
magnetic field in an inductor takes energy that must be supplied by the external electromotive source. This energy is ‘L.I2/2’,
where ‘L’ is the inductance and ‘I’ is the current; the current is simply a count of the moving electrons,
and hence is proportional to the total moving electrostatic field.
In order to make higher inductances the wires are formed in a loop or coil so that the induced magnetic fields
from different electrons overlap and add. The electric fields from electrons in adjacent loops add linearly, so the energy
involved increases as the square. This is why inductance ‘L’ is proportional to the square of the number of turns
in the loop, and is clear proof that the electric field of the electron has inertia.
When current is flowing in a circuit through an inductor, and the circuit is suddenly broken open, a spark
will jump across the break. This is caused by the augmented inertia associated with the electrons’ overlapping electric
fields - they cannot stop dead, but have to dissipate their induced magnetic energy. This is a clear demonstration that an
electron’s inertia owes much to its elecrostatic field.
Adding a ferrous core to such a coil increases the induced magnetic field dramatically, leading to much increased
induced inertia. |